Relations & Functions - Class 12 NCERT | Summary

Relations & Functions — Class 12 NCERT

Concise and exam-smart summary covering definitions, properties, types, composites, inverses and core results.

Overview

The chapter develops the idea of linking elements across sets using relations and special relations called functions. It revisits Cartesian products and uses them to define relations. Key objectives are to identify relation properties, classify functions (injective, surjective, bijective), and work with composition and inverses.

Cartesian Product

If A and B are sets, their Cartesian product is

A × B = { (a, b) : a ∈ A, b ∈ B }

If |A| = m and |B| = n, then |A × B| = m · n.

Relations

A relation R from A to B is a subset of A × B. If A = B, R is a relation on A.

Properties of Relations

Reflexive: (a,a) ∈ R for all a ∈ A.
Symmetric: (a,b) ∈ R ⇒ (b,a) ∈ R.
Transitive: (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R.
Equivalence relation: reflexive + symmetric + transitive. It partitions A into equivalence classes [a] = { x ∈ A : x ~ a }.

Functions (Definition)

A function f: A → B is a relation where every element of A has exactly one image in B. Notation: f(a) = b.

Domain & Range

Domain: set of inputs (A).
Codomain: target set (B).
Range: actual set of outputs { f(a) : a ∈ A }.

Types of Functions

Injective (One–One): f(a1)=f(a2) ⇒ a1=a2.
Surjective (Onto): Range(f) = Codomain.
Bijective: Both injective and surjective — invertible.
Identity: I_A(a)=a.
Constant: f(x)=c for all x.

Composite & Inverse

Composite: If f: A → B and g: B → C, then

(g ∘ f)(x) = g(f(x))

Inverse: f⁻¹ exists iff f is bijective. Then f⁻¹(f(x)) = x and f(f⁻¹(y)) = y.

Common Exam Tasks

Check properties (reflexive/symmetric/transitive) of a given relation.
Find domain, codomain, and range of functions.
Prove one–one or onto using standard templates.
Compute composite functions and test invertibility.

Key Results & Formulas

A × B = { (a,b) }, |A × B| = |A||B|.
Composite associativity: h ∘ (g ∘ f) = (h ∘ g) ∘ f.
Inverse of a bijection is a bijection.
Identity function acts as neutral element: f ∘ I = I ∘ f = f.

Equivalence Classes & Partition

An equivalence relation partitions the set into disjoint equivalence classes. Each element belongs to exactly one class. Useful for counting and modular arithmetic examples.

Worked Strategy (How to Answer)

Write down domain, codomain, and range explicitly.
State what you are proving (e.g., reflexive) before algebraic steps.
Use template proofs: assume f(a1)=f(a2) to show injective; set f(x)=y and solve for x to show onto.
For composite functions show domain compatibility first.

Exam Tips & Common Pitfalls

Always mention the order when working with Cartesian products or relations.
When testing properties, use counterexamples to disprove quickly.
Remember: AB = O does not imply A = O or B = O for relations/compositions. (Be careful with interpretations.)
Practice PYQs: patterns repeat — equivalence relations and invertibility questions are frequent.

📘 Detailed Summary: Relations and Functions – Class 12 NCERT

The chapter Relations and Functions continues from Class 11 and deepens your understanding of how sets interact through mappings. It starts by revisiting Cartesian products, which form the foundation for defining relations. If $A$ and $B$ are two sets, all possible ordered pairs $(a, b)$ form the Cartesian product $\times B$ . A relation is simply any subset of this product.

1. Relations and Their Properties

A relation tells us how elements of one set connect to elements of another (or the same) set. The chapter explores important properties:

Reflexive: Every element relates to itself.
Symmetric: If $a$ is related to $b$ , then $b$ must be related to $a$ .
Transitive: If $a$ relates to $b$ and $b$ relates to $c$ , then $a$ must relate to $c$ .
Equivalence Relation: One that is reflexive + symmetric + transitive. It partitions a set into equivalence classes.

2. Functions and Their Types

A function is a special relation where each element of the domain has exactly one image in the codomain. The chapter describes three important types:

Injective (One–One): Each value in the codomain has at most one pre-image.
Surjective (Onto): The function covers the entire codomain.
Bijective: Both injective and surjective — essential for invertible functions.

Understanding the behavior of functions helps in later topics like calculus and transformations.

3. Composite and Invertible Functions

If functions are compatible (i.e., the range of one fits the domain of another), they can be combined to form composite functions.
A function is invertible only when it is bijective, making its inverse a well-defined function.

This chapter strengthens the conceptual backbone for later chapters on continuity, differentiation, and advanced algebra.

📌 Key Formulas & Definitions

1. Cartesian Product

$\times B = \{(a, b) : a \in A,\, b \in B\}$

If $∣ A ∣ = m$ and $∣ B ∣ = n$ , then:

$\times B| = mn$

2. Relation

A relation $R$ from $A$ to $B$ is:

$\subseteq A \times B$

3. Function

A function $\to B$ is a relation where every $\in A$ has exactly one image $\in B$ .

4. Types of Functions

One–One (Injective):

$f(a1)=f(a2)⇒a1=a2f(a_1) = f(a_2) \Rightarrow a_1 = a_2$

Onto (Surjective):

$Range(f)=Codomain\text{Range}(f) = \text{Codomain}$

Bijective:
Both one–one and onto.

5. Composite Function

If $\to B$ and $\to C$ , then the composite:

$\circ f)(x) = g(f(x))$

6. Invertible Function

A function $f$ is invertible iff it is bijective.

Inverse satisfies:

$f^{-1}(f(x)) = x$

and

$f(f^{-1}(y)) = y$

7. Identity Function

$IA(a)=a∀a∈AI_A(a) = a \quad \forall a \in A$

8. Constant Function

$\quad (\text{for some constant } k)$

🔥 Tips to Score High by Practicing Previous Year Questions (PYQs) of This Chapter

1️⃣ Start by spotting repeating patterns

PYQs for this chapter are weirdly predictable.
You’ll see the same ideas pop up again and again —

Check if a given relation is reflexive/symmetric/transitive
Find if a function is one–one/onto
Compute composite function
Find inverse of a bijection

If a concept showed up twice in the last 10 years, trust me, it’s worth mastering.

2️⃣ Practice each type until it feels mechanical

Don’t just “solve” the question — solve one type 5–6 times.
When your brain starts going: “Oh, reflexive? Easy, check diagonal pairs.”
…you’ve unlocked exam mode.

3️⃣ Mark your weak spots and focus only on those

Still confused about transitivity proofs?
Composite functions tripping you up?
Don’t waste time re-solving what you already nailed.
Double-down on the 20% that trips you — that’s where marks hide.

4️⃣ Time yourself like you’re in the actual exam

This chapter gives HIGH chance questions in Section A/B (short questions).
Practicing with a timer helps because:

You stop overthinking
Your solution steps become compact
You don’t panic when you see long relation definitions in the exam

5️⃣ Learn the “default approach” for each type of problem

For example:

To check one–one: assume $f(a_1)=f(a_2)$ → show $a_1=a_2$ .
To check onto: let output = $y$ → solve for $x$ .
Composite function: apply inside first, outside next.
When you follow a fixed template, you avoid silly step errors.

6️⃣ Write definitions in your own words — and memorize them cleanly

Examiners LOVE asking definitions in short questions.
If you can write reflexive/symmetric/transitive/one–one/onto crisply,
that’s free marks.

7️⃣ Focus on presentation — this chapter is VERY step-sensitive

Half the marks in functions problems come from:

Writing domain/codomain
Showing the equation clearly
Stating final yes/no with reason
Clean steps = clean marks.

8️⃣ Re-solve at least 5 PYQs on equivalence relations

This part is a scoring jackpot.
If you understand “reflexive + symmetric + transitive,”
you’ve basically secured 4–6 marks automatically.

9️⃣ Keep a quick revision sheet

Just one page with:

Definitions
Types of functions
Formulas
Steps to check properties
Glance at it once before solving PYQs. Your speed will double.

🔟 Revisit your wrong answers after 24 hours

This locks the concept in your memory.
Don’t just read the correction — redo it from scratch. That’s how toppers do it

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